# Questions tagged [options]

A contract that gives the owner the right, but not the obligation, to buy or sell a security at a fixed price in the future.

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### Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
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Given, a numéraire $(N(t))_{0\leq t \leq T}$ and an index $(X(t))_{0\leq t\leq T}$ that is a $\mathbb Q^{N}$-martingale, we consider the natural payoff $V_{N}(T)$, where it pays $$V_{N}(T):=X(T)N(T) \... 2 votes 0 answers 26 views ### Finding the distribution of I(T_{1},T_{n}) under an appropriate measure if the forwards are lognormal? [duplicate] My question follows beneath the "lengthy" setting I describe: Given a tenor discretization 0 = T_{0}< ... < T_{n} =T, and under the assumption that under \mathbb P, for all i = 1,.... -1 votes 2 answers 326 views ### Easier way than using QuantLib to compute the price and Greeks of a vanilla European option? I'm using the following to compute the price and Greeks a vanilla European option: ... • 389 0 votes 3 answers 790 views ### Early exercising American put options I have found a proof that an American put option without dividend will never be exercised early. However, I suspect that that is not true, so there should be a mistake in the proof. The proof is as ... • 103 7 votes 3 answers 721 views ### Why does the diffusion term remain the same when we change pricing measure? Consider some Itô process dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t} under the measure \mathbb P, where W^{\mathbb P} is a \mathbb P-Brownian motion In plenty of interest rate examples, I have ... 1 vote 1 answer 138 views ### implied vol by Delta I am looking at some data that is Delta 10, Delta 30, etc for an index option CDX IG. I know the meaning of Delta, as a sensitivity of the price move with respect 1 move in the underlying index. What ... • 447 1 vote 0 answers 82 views ### How To Construct A Volatility Spread Position? Is there a simple way to spread the volatility of one product against another? By simple I mean one trade executed on each leg rather than constant delta hedging. I can see a lot of opportunity for ... • 11 0 votes 1 answer 94 views ### Why would valuation for a swap be the same on the backward and forward rate but not a caplet Consider for time discretization 0 = T_{0} < T_{1} <... < S < T < T_{n}, and the corresponding forward rates and backward rate: \text{Forward rate: }L(S,T;t) \text{Backward Rate: }... 2 votes 3 answers 384 views ### Backtesting Option Strategies with IV Data Only I’ve tried to find a good answer for this but had no luck so I’m bringing it here: potentially beginner question, but how much accuracy would I be sacrificing by backtesting an options strategy with ... 0 votes 1 answer 97 views ### Is there a closed form formula for the value of a European Put KO/KI? Was able to find closed form formula for single barrier options KO OR KI. However I haven't found that for a double barrier option. I am looking for a put down & in KI, up and out KO, where: H(KI) ... 1 vote 0 answers 295 views ### How does (d_2/\sigma) = (1-d_1) while deriving the Vanna Formula from BSM? [closed] Just realized there was a quant finance board, so I figured I'd post it here instead. I'm trying to derive Vanna from the Black-Scholes Model (BSM) equation, but had a hook up on one of the ... 1 vote 0 answers 70 views ### Why is the parity graph in Natenberg shifted up? In chapter 4 of Natenberg's "Option and Volatility and pricing", he discusses how to draw parity graphs for option positions. These are defined as a plot of the intrinsic value of the ... • 151 1 vote 0 answers 43 views ### Assymetric Rate Distribution The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ... • 5,517 1 vote 0 answers 256 views ### at what frequency do option market makers delta hedge Could someone with option market making experience tell me usually at what frequency do the major option market makers delta-hedge their positions (say for US single stocks or equity indices)? ... • 271 1 vote 0 answers 39 views ### How to find the risk neutral valuation of P(T_{1}) und the measure \mathbb Q^{P(T_{2})} How do I find the risk neutral valuation of P(T_{1}) und the measure \mathbb Q^{P(T_{2})}, where P(T_{1}) and P(T_{2}) refer to the T_{1} and T_{2} zero coupon bond with 0 < T_{1} < ... 1 vote 0 answers 72 views ### What adjustments need to be made before a Monte-Carlo simulation can be applied for the exotic option (L_{\text{domestic}}-L_{\text{foreign}})^{+} I just want to reassure myself that I understand why Monte-Carlo is the appropriate tool in computing the fair value prices for different options. Let's say we have a Tenor discretization T_{0}=0<... 1 vote 3 answers 463 views ### Gamma PnL when hedging with implied volatility - where is the mark to market PnL? It is well known that hedging with implied volatility involves a PnL: 0.5*(σ^{2}_r−σ^{2}_i)S^{2}*Γ_{i}dt In the Wilmott paper (http://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch.pdf), they imply ... • 123 0 votes 1 answer 117 views ### Understanding the expected value of the average I've been looking into Asian Options pricing. Part of the process is about looking for the expected value of the average of a time series undergoing e.g. geometric brownian motion. I came across this ... 7 votes 0 answers 130 views ### Implied vol bounded if and only if instantaneous vol bounded I'd like to show that in diffusion models IV is bounded iff instantaneous vol is bounded if there is to be no arbitrage. So, assume a model under the pricing measure of the form$$ dS_u = \sigma_u S_u ... 1 vote
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### Issues with calculating IV with options bar data

I am currently working with some options OHLC data (30 minute bars) from IBKR for a range of strike prices, maturities and for both calls/puts. For each bar, I am trying to back out the IV (crudely ...
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### Under put call parity shouldnt the implied volatility for call and put for same strike and maturity be the same?

If all of the other inputs into black scholes (divs/rates/time to maturity/strick/current price/etc) are all the same between two pairs of calls/put contracts on the same security, shouldn't the ...
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### In FX markets, option can be expressed as either call or put. Explain

For example, if option contract has condition: $AUDUSD = 0.8$ at the maturity date, and current exchange rate is $1 AUD = 0.75 USD$. For this option, it could be considered a call option on $USD$, and ...
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### Valuation of chooser options

The below formula for valuation of chooser options from Hull's book is not making sense to me. Why do we use call value at time T=0 while we use put value using t=0 call value and discount strike and ...
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### Confusion with the equity option skew

In general out of the money (OTM) equity options have higher implied volatility (IV) than at the money (ATM) options. So assuming we have two put options (5% OTM and 10% OTM). Skew reveals that 10% ...
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### Is there a financial instrument that is exposed to the change in growth of an asset over time?

Is there a financial instrument that is exposed to the rate of change of the value of a specific asset? If I believe a stock price will continue to grow in the future, but grow more slowly than in the ...
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### Numerical scheme for this HJB equation

Without dwelling on details on how to obtain the HJB equation for this problem, I would like to know if the scheme I wrote for solving it numerically is viable or did I miss something. I need to solve ...
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### Determine Strikes on Option Chain

Does anyone know how to determine option strikes on an option chain are determined for a specific stock? I have been searching online and can't seem to figure out how/why the specific strike are set. ...
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### Volatility of American vs European Stock option return

Let's say that I hold an American Call Option (ACO) and an European Call Option (ECO) in my portfolio on the same underlying, with same strike price and same maturity date. Given that I hold both ...
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### Delta of FX Options, Different Currency in Trading Book - Trading Interview Question

Having done stochastic analysis in university, together with tons of other math courses, do never prepare you for an actual interview in trading. Stumbled on what I believe might be an easy question, ...
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### Monte Carlo: How to interpolate Dupire's Local Volatility

I am trying to price barrier options which can have daily or monthly observations. I first calibrated by Black vols into smooth SVI vols (with linear interpolation along time in variance) to obtain ...
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### Calculate options prices based on given options and spread prices

Suppose you know the following information: Futures price on a stock is 66 70 strike straddle is trading at 27 50-60 put spread is trading at 2.5 50-60-70 put butterfly is trading at 0.2 Assume ...
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### Evaluating swaptions with negative interest rates

Does anyone know if it is possible to evaluate swaptions with negative interest rates with Quantlib? ...
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### Is it true that interest rates options with different maturities are free of calendar arbitrage because of the different underlying rates dynamics?

The title says it all - is it true that European style interest rates options (lets say on LIBOR 3M for the sake of simplicity) with different maturities are free of calendar arbitrage because ...
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### Black Scholes derivation: Why treat Delta as a constant?

In the derivation of the Black-Scholes equation, it is argued (e.g. in the original paper and in Hull) that $$dV(S_t, t)=(…)dt + \frac{\partial V}{\partial S} dS_t,$$ where $V(S_t, t)$ is the value at ...
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### Purpose of Vega Hedging

I am trying to understand the principle of vega hedging. When should a market maker vega hedge his position ? Let's suppose that a market maker delta and gamma hedge himself, and carries his position (...
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### Optimize call option purchase

If it is predicted that the price of a stock will increase from P1 to between P2 and P3 in time T (assume the distribution of the price will be evenly distributed between the range of [P2, P3] at time ...
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### Distribution of total delta of option portfolio

We know the delta of a portfolio of options is simply the sum of deltas of the individual options. But are there any additional known properties about the total delta (or other greeks) of a portfolio ...
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### No-arbitrage conditions on a caps/floors volatility surface

Suppose that one has a caps/floors volatility surface and wants to check whether this surface admits arbitrage. What is the theoretical and practical way to do it? Lets talk only about caps for ...
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### Why can’t delta’s be used to price double no touch options?

Here is the link to a MATLAB one touch option pricing calculator I used:OT I tried several inputs and I noticed that the one touch option price is approximately twice the delta of an equivalent ...
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### Find the value of put option using a two-period binomial model

I've been asked to find the price of a two-month European Put Option with strike price $£40$. The price at $S_0=£30$, this can move up to $£40$ or down to $£25$ ($1/3$ chance to go up, $2/3$ chance to ...
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### Power options for pricing European claims

I have the following question: Why would somebody be interested in the expression $E[S^\theta]$ for $\theta$ between zero and one. The only thing I know is that this then can be somehow used to ...
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### Approximating Volatility Skew From historic returns? [closed]

I was wondering if someone could help me with something. I've been reading more about equity options, and I'm struggling with skew. Conceptually I understand why it exists, what I'm struggling with is ...
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### Generating Greeks with American Options

Investor and Software Engineer but very new to quant finance here... I have the below code (which I'm sure will be helpful for some) and have some questions regarding the function parameters! Is RF ...
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### No Probability in Greeks

In an interview, I was once told that I should not consider probability when talking about option greeks since from a mathematical point of view greeks have nothing to do with probability. That is of ...
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### Option pricing with risk-neutral approach

Problem Given $Y_t$ price of a stock (no-dividents), and a derivative paying $Y_T^2$ at maturity $T$, evaluate the price of the instrument now using risk-neutral approach and check that it satisfies ...
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### Easy, but doubts - Annualize daily turnover

I am fairly certain I am correct but I just want to double-check on portfolio turnover calculation. I need to annualize the daily turnover rate. To calculate, the daily turnover, I am using the ...
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### Greeks of portfolio in response to underlying price change

I'm trying to wrap my head around Greeks, and I'm getting a little bit confused. For example, let's say my portfolio holds a long 5 month ATM call with strike \\$20, and short 2 month OTM call with ...
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### CDS Option Pricing (Missing Index Factor)

I've read the OpenGamma paper https://quant.opengamma.io/CDS-Options-OpenGamma.pdf on CDS Options, and noticed a small discrepancy. So I wanted to double-check my understanding. In Section 6.4 the ...
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